I would like to optimize a function of the form $$u(x) = \begin{cases} \exp{(-\lambda x)} & \text{if } a \le x \le b \\ exp(-\lambda b) & \text{if } x > b \\ -\infty &\text{if } x< a\end{cases}$$ for some real constants $a,b$. I'm not very experienced in optimization and would like to know if this is feasible or how one needs to tweak this function to make it more tractable. Reference or complete answers are very appreciated.
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You need to define whether you are minimizing or maximizing, and with respect to which variables, as well as whether there are any constraints on those variable(s). If you are minimizing with respect to x and x is unconstrained, the optimum value for x is any x < a, and the optimal objective value is $-\infty$ – Mark L. Stone Jun 20 '17 at 23:01